The former Prime Minister said it would be a "good idea if Mr Cameron took on the SNP leader in the run-up to September's referendum.
Mr Brown also repeated his warning that pro-union politicians were allowing nationalists to portray the vote as "England against Scotland".
Speaking to journalists at a Westminster lunch, the MP dismissed arguments that Mr Cameron should stand back from the battle because he was toxic to Scottish voters.
"The Prime Minister has got to be part of this debate," he said.
Asked whether he thought that meant Mr Cameron should agree to the Scottish First Minister's demands for a debate, Mr Brown replied: "People have to decide what the best way forward is.
"I think it would be a good idea if David Cameron did debate Alex Salmond, but I'm not involved in the negotiations."
Firing salvos at both sides as he waded into the independence fight 100 days before Scots go to the polls, Mr Brown condemned Mr Salmond as "embarrassing".
He said the First Minister had made making Scotland look "small" by unfurling a Saltire flag behind Mr Cameron when Andy Murray won Wimbledon last year.
But Mr Brown also criticised the way Scots had been warned that they would not be allowed to share the Pound if they vote Yes.
He complained about "patronising" government publicity material suggesting Scots would be able to buy fish and chips every day for 10 weeks with the money they save by voting 'No' to independence.
"Countries can be lost by mistake... It must be two visions of Scotland's future that are at stake."
Mr Brown made light of brickbats about his lack of attendance at Westminster since leaving Downing Street in 2010, joking that he was going to "hire a tour guide" to find his way around.
But he declined to confirm that he will be staying as MP for Kirkcaldy and Cowdenbeath after the general election, saying only: "I am not going to make any announcements while we are fighting a referendum."
Mr Brown played down the importance of immigration and Euroscepticism in Ukip's recent electoral gains, saying the underlying problem was the economy and globalisation.
"When you look at the state of the British economy what people are actually looking for is better jobs," he said.
In a swipe at Ukip leader Nigel Farage, Mr Brown joked: "Britain wouldn't be Britain without Farage with a pint in one hand and a mystery lady in the other."
SNP MSP Annabelle Ewing said: "The fact that even Gordon Brown recognises the need for a First Minister to Prime Minister debate speaks volumes - and will be a source of huge embarrassment for Alistair Darling, who just this week claimed that he was the man to debate the First Minister.
"Alistair Darling will be used to being undermined by his Tory colleagues in the No campaign - but now he's being undermined by a former Labour Prime Minister.
"The pressure is now building on David Cameron to finally agree to the head to head debate the people of Scotland want to see - and he is fast running out of places to hide."
Downing Street sidestepped the calls for Mr Cameron to go head to head with the First Minister.
The Prime Minister's official spokesman said: "I think the Prime Minister's view is the same as the Better Together campaign, which is they very much look forward to Alex Salmond agreeing to debate Alistair Darling."